Problem 37 - Algebra 2 and Trigonometry

Question

In \(26-37,\) find each product. $$ \left(-\frac{1}{8}+\frac{1}{8} i\right)(4+4 i) $$

Step-by-Step Solution

Answer

The product is \(-1\).

Step 1: Recognize the complex numbers

Identify the complex numbers involved in the expression. The first complex number is \(-\frac{1}{8}+\frac{1}{8}i\) and the second complex number is \(4+4i\).

Step 2: Apply the distributive property

Use the distributive property to multiply the complex numbers: \[\left(-\frac{1}{8}\right)(4) + \left(-\frac{1}{8}\right)(4i) + \left(\frac{1}{8}i\right)(4) + \left(\frac{1}{8}i\right)(4i)\]

Step 3: Perform the multiplications

Calculate each part separately:- \(-\frac{1}{8} \times 4 = -\frac{4}{8} = -\frac{1}{2}\)- \(-\frac{1}{8} \times 4i = -\frac{4}{8}i = -\frac{1}{2}i\)- \(\frac{1}{8}i \times 4 = \frac{4}{8}i = \frac{1}{2}i\)- \(\frac{1}{8}i \times 4i = \frac{4}{8}i^2 = \frac{1}{2}i^2 = \frac{1}{2}(-1) = -\frac{1}{2}\)

Step 4: Combine like terms

Combine the real parts and the imaginary parts separately:- Real part: \(-\frac{1}{2} - \frac{1}{2} = -1\)- Imaginary part: \(-\frac{1}{2}i + \frac{1}{2}i = 0\i\)Thus, the expression simplifies to \(-1 + 0i = -1\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics, frequently used in algebraic operations. It involves the rule that states you can distribute multiplication across addition or subtraction within an expression. This property is often written as:

• For any numbers or variables \(a, b, \text{ and } c\): \((a + b)c = ac + bc\)

In the context of multiplying complex numbers, like in our exercise, the distributive property helps to systematically expand expressions. Given two complex numbers, say \(a + bi\) and \(c + di\), you distribute each term from the first complex number across each term from the second complex number.

• This means applying: \( (a + bi)(c + di) = ac + adi + bci + bdi^2\).

Remember, every term from the first expression interacts with every term in the second, ensuring all combinations are covered.

Imaginary Unit

The imaginary unit, denoted as \i\, is a unique mathematical concept used to expand the real number system into the complex numbers. Defined by the equation:

• \i^2 = -1\

The need for the imaginary unit arises when considering numbers that yield negative values when squared, a scenario not possible with real numbers. The inclusion of \i\ allows for the representation of numbers in the form \(a + bi\), where \a\ and \b\ are real numbers.
The imaginary unit plays a critical role in various fields of mathematics and engineering, especially in electrical engineering for handling sinusoidal functions related to alternating current.
In this exercise, the term \(i\) is used as part of the computation, especially when you multiply two imaginary components, yielding \(i^2\), which simplifies to \-1\.

Multiplying Complex Numbers

Multiplying complex numbers might seem daunting at first, but it's quite simple once you get the hang of it, especially when using the distributive property. When multiplying two complex numbers, \(a + bi\) and \(c + di\), follow these steps:

• First, distribute each term in the first complex number across each term in the second complex number.
• Compute the real parts: \(ac\) and imaginary parts: \(ad\), \(bc\), and \(bd\).
• Remember, multiplying \(bdi\) yields \(bd i^2 = bd(-1)\) due to \(i^2 = -1\).

Finally, combine like terms:

• Add all real parts together.
• Combine all imaginary parts.

Using these steps with the given numbers in the exercise, helps simplify the product to a real number. Ultimately, \(-1\) in this particular case, since the imaginary components cancel out, demonstrating a common scenario when multiplying complex conjugates.